RD Sharma solutions for Mathematics [English] Class 10 chapter 4 - Quadratic Equations [Latest edition]

Check whether the following is quadratic equation or not.

x² + 6x − 4 = 0

Check whether the following is quadratic equation or not.

`sqrt(3x^2)-2x+1/2=0`

Check whether the following is quadratic equation or not.

`x^2+1/x^2=5`

Check whether the following is quadratic equation or not.

`x-3/x=x^2`

Check whether the following is quadratic equation or not.

`2x^2-sqrt(3x)+9=0`

Check whether the following is quadratic equation or not.

`x^2 - 2x - sqrtx - 5 = 0`

Check whether the following is quadratic equation or not.

3x² - 5x + 9 = x² - 7x + 3

Check whether the following is quadratic equation or not.

`x+1/x=1`

Check whether the following is quadratic equation or not.

x² - 3x = 0

Check whether the following is quadratic equation or not.

`(x+1/x)^2=3(1+1/x)+4`

Check whether the following is quadratic equation or not.

(2𝑥 + 1)(3𝑥 + 2) = 6(𝑥 − 1)(𝑥 − 2)

Check whether the following is quadratic equation or not.

`x+1/x=x^2`, x ≠ 0

Check whether the following is quadratic equation or not.

16x² − 3 = (2x + 5) (5x − 3)

Check whether the following is quadratic equation or not.

(x + 2)³ = x³ − 4

Check whether the following is quadratic equation or not.

x(x + 1) + 8 = (x + 2) (x - 2)

In the following, determine whether the given values are solutions of the given equation or not:

x² - 3x + 2 = 0, x = 2, x = -1

In the following, determine whether the given values are solutions of the given equation or not:

x² + x + 1 = 0, x = 0, x = 1

In the following, determine whether the given values are solutions of the given equation or not:

`x^2 - 3sqrt3x+6=0`, `x=sqrt3`, `x=-2sqrt3`

In the following, determine whether the given values are solutions of the given equation or not:

`x+1/x=13/6`, `x=5/6`, `x=4/3`

In the following, determine whether the given values are solutions of the given equation or not:

2x² - x + 9 = x² + 4x + 3, x = 2, x =3

In the following, determine whether the given values are solutions of the given equation or not:

`x^2-sqrt2x-4=0`, `x=-sqrt2`, `x=-2sqrt2`

In the following, determine whether the given values are solutions of the given equation or not:

a²x² - 3abx + 2b² = 0, `x=a/b`, `x=b/a`

In the following, find the value of k for which the given value is a solution of the given equation:

7x² + kx - 3 = 0, `x=2/3`

In the following, find the value of k for which the given value is a solution of the given equation:

x² - x(a + b) + k = 0, x = a

In the following, find the value of k for which the given value is a solution of the given equation:

`kx^2+sqrt2x-4=0`, `x=sqrt2`

In the following, find the value of k for which the given value is a solution of the given equation:

x² + 3ax + k = 0, x = -a

Determine if, 3 is a root of the equation given below:

`sqrt(x^2-4x+3)+sqrt(x^2-9)=sqrt(4x^2-14x+16)`

If x = 2/3 and x = −3 are the roots of the equation ax² + 7x + b = 0, find the values of aand b.

The product of two consecutive positive integers is 306. Form the quadratic equation to find the integers, if x denotes the smaller integer.

John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 128. Form the quadratic equation to find how many marbles they had to start with, if John had x marbles.

A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of articles produced in a day. On a particular day, the total cost of production was Rs. 750. If x denotes the number of toys produced that day, form the quadratic equation fo find x. 

The height of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, form the quadratic equation to find the base of the triangle.

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore. If the average speed of the express train is 1 1 km/hr more than that of the passenger train, form the quadratic equation to find the average speed of express train.

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Solve the following quadratic equations by factorization:

(x − 4) (x + 2) = 0

Solve the following quadratic equations by factorization:

(2x + 3)(3x − 7) = 0

Solve the following quadratic equations by factorization:

3x² − 14x − 5 = 0

Solve the following quadratic equations by factorization:

9x² − 3x − 2 = 0

Solve the following quadratic equations by factorization:

`1/(x-1)-1/(x+5)=6/7` , x ≠ 1, -5

Solve the following quadratic equations by factorization:

6x² + 11x + 3 = 0

Solve the following quadratic equations by factorization:

5x² - 3x - 2 = 0

Solve the following quadratic equations by factorization:

48x² − 13x − 1 = 0

Solve the following quadratic equations by factorization:

3x² = -11x - 10

Solve the following quadratic equations by factorization:

25x(x + 1) = -4

Solve the following quadratic equations by factorization:

\[16x - \frac{10}{x} = 27\]

Solve the following quadratic equations by factorization:

`1/x-1/(x-2)=3` , x ≠ 0, 2

Solve the following quadratic equations by factorization:

`1/(x+4)-1/(x-7)=11/30` , x ≠ 4, 7

Solve the following quadratic equations by factorization:\[\frac{1}{x - 3} + \frac{2}{x - 2} = \frac{8}{x}; x \neq 0, 2, 3\]

Solve the following quadratic equations by factorization:

a²x² - 3abx + 2b² = 0

Solve the following quadratic equations by factorization:

\[9 x^2 - 6 b^2 x - \left( a^4 - b^4 \right) = 0\]

Solve the following quadratic equations by factorization:

4x² + 4bx - (a² - b²) = 0

Solve the following quadratic equations by factorization:

ax² + (4a² - 3b)x - 12ab = 0

Solve the following quadratic equations by factorization: \[2 x^2 + ax - a^2 = 0\]

Solve the following quadratic equations by factorization: \[\frac{16}{x} - 1 = \frac{15}{x + 1}; x \neq 0, - 1\]

Solve the following quadratic equations by factorization:

`(x+3)/(x+2)=(3x-7)/(2x-3)`

Solve the following quadratic equations by factorization:

`(2x)/(x-4)+(2x-5)/(x-3)=25/3`

Solve the following quadratic equations by factorization:

`(x+3)/(x-2)-(1-x)/x=17/4`

Solve the following quadratic equations by factorization:

`(x-3)/(x+3)-(x+3)/(x-3)=48/7` , x ≠ 3, x ≠ -3

Solve the following quadratic equations by factorization:

`1/(x-2)+2/(x-1)=6/x` , x ≠ 0

Solve the following quadratic equations by factorization:

`(x+1)/(x-1)-(x-1)/(x+1)=5/6` , x ≠ 1, x ≠ -1

Solve the following quadratic equations by factorization:

`(x-1)/(2x+1)+(2x+1)/(x-1)=5/2` , x ≠ -1/2, 1

Solve the following quadratic equations by factorization:

\[\frac{4}{x} - 3 = \frac{5}{2x + 3}, x \neq 0, - \frac{3}{2}\]

Solve the following quadratic equations by factorization: \[\frac{x - 4}{x - 5} + \frac{x - 6}{x - 7} = \frac{10}{3}; x \neq 5, 7\]

Solve the following quadratic equations by factorization:

\[\frac{x - 2}{x - 3} + \frac{x - 4}{x - 5} = \frac{10}{3}; x \neq 3, 5\]

Solve the following quadratic equations by factorization: \[\frac{5 + x}{5 - x} - \frac{5 - x}{5 + x} = 3\frac{3}{4}; x \neq 5, - 5\]

Solve the following quadratic equations by factorization: \[\frac{3}{x + 1} - \frac{1}{2} = \frac{2}{3x - 1}, x \neq - 1, \frac{1}{3}\]

Solve the following quadratic equations by factorization: \[\frac{3}{x + 1} + \frac{4}{x - 1} = \frac{29}{4x - 1}; x \neq 1, - 1, \frac{1}{4}\]

Solve the following quadratic equations by factorization: \[\frac{2}{x + 1} + \frac{3}{2(x - 2)} = \frac{23}{5x}; x \neq 0, - 1, 2\]

Solve the following quadratic equations by factorization: \[\sqrt{3} x^2 - 2\sqrt{2}x - 2\sqrt{3} = 0\]

Solve the following quadratic equations by factorization:

`4sqrt3x^2+5x-2sqrt3=0`

Solve the following quadratic equations by factorization:

`sqrt2x^2-3x-2sqrt2=0`

Solve the following quadratic equations by factorization:

`3x^2-2sqrt6x+2=0`

Find the roots of the quadratic equation \[\sqrt{2} x^2 + 7x + 5\sqrt{2} = 0\].

Solve the following quadratic equations by factorization:

`m/nx^2+n/m=1-2x`

Solve the following quadratic equations by factorization:

`(x-a)/(x-b)+(x-b)/(x-a)=a/b+b/a`

Solve the following quadratic equations by factorization:

`1/((x-1)(x-2))+1/((x-2)(x-3))+1/((x-3)(x-4))=1/6`

Solve the following quadratic equation by factorization: \[\frac{a}{x - b} + \frac{b}{x - a} = 2\]

Solve the following quadratic equations by factorization: \[\frac{x + 1}{x - 1} + \frac{x - 2}{x + 2} = 4 - \frac{2x + 3}{x - 2};   x \neq 1,  - 2,   2\] 

Solve the following quadratic equations by factorization:

`a/(x-a)+b/(x-b)=(2c)/(x-c)`

Solve the following quadratic equations by factorization:

x² + 2ab = (2a + b)x

Solve the following quadratic equations by factorization:

(a + b)²x² - 4abx - (a - b)² = 0

Solve the following quadratic equations by factorization:

a(x² + 1) - x(a² + 1) = 0

Solve the following quadratic equations by factorization:

x² - x - a(a + 1) = 0

Solve the following quadratic equations by factorization:

`x^2+(a+1/a)x+1=0`

Solve the following quadratic equations by factorization:

abx² + (b² – ac)x – bc = 0

Solve the following quadratic equations by factorization:

a²b²x² + b²x - a²x - 1 = 0

Solve the following quadratic equations by factorization:

`(x-1)/(x-2)+(x-3)/(x-4)=3 1/3`, x ≠ 2, 4

Solve the following quadratic equations by factorization: \[\frac{1}{2a + b + 2x} = \frac{1}{2a} + \frac{1}{b} + \frac{1}{2x}\]

Solve the following quadratic equations by factorization:

\[3\left( \frac{3x - 1}{2x + 3} \right) - 2\left( \frac{2x + 3}{3x - 1} \right) = 5; x \neq \frac{1}{3}, - \frac{3}{2}\]

Solve the following quadratic equations by factorization:

\[3\left( \frac{7x + 1}{5x - 3} \right) - 4\left( \frac{5x - 3}{7x + 1} \right) = 11; x \neq \frac{3}{5}, - \frac{1}{7}\]

Solve the following quadratic equation by factorization:

`(x-5)(x-6)=25/(24)^2`

Solve the following quadratic equations by factorization:

`7x + 3/x=35 3/5`

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

`x^2-4sqrt2x+6=0`

Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x² – 7x + 3  = 0

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

3x² + 11x + 10 = 0

Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x² + x – 4 =  0

Find the roots of the following quadratic equations, if they exist, by the method of completing the square 2x² + x + 4 = 0

Find the roots of the following quadratic equations, if they exist, by the method of completing the square `4x^2 + 4sqrt3x + 3 = 0`

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

`sqrt2x^2-3x-2sqrt2=0`

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

`sqrt3x^2+10x+7sqrt3=0`

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

`x^2-(sqrt2+1)x+sqrt2=0`

Find the roots of the following quadratic equations (if they exist) by the method of completing the square.

x² - 4ax + 4a² - b² = 0

Write the discriminant of the following quadratic equations:

2x² - 5x + 3 = 0

Write the discriminant of the following quadratic equations:

x² + 2x + 4 = 0

Write the discriminant of the following quadratic equations:

(x − 1) (2x − 1) = 0

Write the discriminant of the following quadratic equations:

x² - 2x + k = 0, k ∈ R

Write the discriminant of the following quadratic equations:

`sqrt3x^2+2sqrt2x-2sqrt3=0`

Write the discriminant of the following quadratic equations:

x² - x + 1 = 0

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

16x² = 24x + 1

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

x² + x + 2 = 0

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

`sqrt3x^2+10x-8sqrt3=0`

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

3x² - 2x + 2 = 0

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

`2x^2-2sqrt6x+3=0`

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

3a²x² + 8abx + 4b² = 0, a ≠ 0

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

`3x^2+2sqrt5x-5=0`

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

x² - 2x + 1 = 0

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

`2x^2+5sqrt3x+6=0`

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

`sqrt2x^2+7x+5sqrt2=0`

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

`2x^2-2sqrt2x+1=0`

In the following, determine whether the given quadratic equation have real roots and if so, find the roots:

3x² - 5x + 2 = 0

Solve for x

`(x-1)/(x-2)+(x-3)/(x-4)=3 1/3`; x ≠ 2, 4

Solve for x:

`1/x - 1/(x-2)=3`, x ≠ 0, 2

Solve for x

`x+1/x=3`, x ≠ 0

Solve for x: \[\frac{16}{x} - 1 = \frac{15}{x + 1}, x \neq 0, - 1\]

Solve for x: \[\frac{1}{x - 3} - \frac{1}{x + 5} = \frac{1}{6}, x \neq 3, - 5\]

Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:

2x² - 3x + 5 = 0

Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:

2x² - 6x + 3 = 0

Determine the nature of the roots of the following quadratic equation:

`3/5x^2-2/3x+1=0`

Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:

`3x^2 - 4sqrt3x + 4 = 0`

Determine the nature of the roots of the following quadratic equation:

`3x^2-2sqrt6x+2=0`

Find the values of k for which the roots are real and equal in each of the following equation:

kx² + 4x + 1 = 0

Find the values of k for which the roots are real and equal in each of the following equation:

`kx^2-2sqrt5x+4=0`

Find the value of k for which the roots are real and equal in the following equation:

3x2 − 5x + 2k = 0

Find the values of k for which the roots are real and equal in each of the following equation:

4x² + kx + 9 = 0

Find the values of k for which the roots are real and equal in each of the following equation:

2kx² - 40x + 25 = 0

Find the values of k for which the roots are real and equal in each of the following equation:

9x² - 24x + k = 0

Find the values of k for which the roots are real and equal in each of the following equation:

4x² - 3kx + 1 = 0

Find the values of k for which the roots are real and equal in each of the following equation:

x² - 2(5 + 2k)x + 3(7 + 10k) = 0

Find the values of k for which the roots are real and equal in each of the following equation:

(3k+1)x² + 2(k + 1)x + k = 0

Find the values of k for which the roots are real and equal in each of the following equation:

kx² + kx + 1 = -4x² - x

Find the values of k for which the roots are real and equal in each of the following equation:

(k + 1)x² + 2(k + 3)x + (k + 8) = 0

Find the values of k for which the roots are real and equal in each of the following equation:

x² - 2kx + 7k - 12 = 0

Find the values of k for which the roots are real and equal in each of the following equation:

(k + 1)x² - 2(3k + 1)x + 8k + 1 = 0

Find the values of k for which the roots are real and equal in each of the following equation:

(2k + 1)x² + 2(k + 3)x + (k + 5) = 0

Find the values of k for which the roots are real and equal in each of the following equation:

4x² - 2(k + 1)x + (k + 4) = 0

Find the values of k for which the roots are real and equal in each of the following equation:

\[4 x^2 - 2\left( k + 1 \right)x + \left( k + 1 \right) = 0\]

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x² + 3x + k = 0

In the following determine the set of values of k for which the given quadratic equation has real roots: \[2 x^2 + x + k = 0\]

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x² - 5x - k = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

kx² + 6x + 1 = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

3x² + 2x + k = 0

Find the value of k for which the following equations have real and equal roots:

\[x^2 - 2\left( k + 1 \right)x + k^2 = 0\]

Find the values of k for which the roots are real and equal in each of the following equation:

k²x² - 2(2k - 1)x + 4 = 0

Find the value of k for which the following equations have real and equal roots:

\[\left( k + 1 \right) x^2 - 2\left( k - 1 \right)x + 1 = 0\]

Find the value of k for which the following equations have real and equal roots:

\[x^2 + k\left( 2x + k - 1 \right) + 2 = 0\]

Find the values of k for which the roots are real and equal in each of the following equation:

2x² + kx + 3 = 0

Find the values of k for which the roots are real and equal in each of the following equation:

kx(x - 2) + 6 = 0

Find the values of k for which the roots are real and equal in each of the following equation:

x² - 4kx + k = 0

Find the values of k for which the roots are real and equal in each of the following equation:

\[kx\left( x - 2\sqrt{5} \right) + 10 = 0\]

Find the values of k for which the roots are real and equal in each of the following equation:\[px(x - 3) + 9 = 0\]

Find the values of k for which the roots are real and equal in each of the following equation:

\[4 x^2 + px + 3 = 0\]

Find the values of k for which the given quadratic equation has real and distinct roots:

kx² + 2x + 1 = 0

Find the values of k for which the given quadratic equation has real and distinct roots:

kx² + 6x + 1 = 0

Find the values of k for which the given quadratic equation has real and distinct roots:

x² - kx + 9 = 0

For what value of k,  (4 - k)x² + (2k + 4)x + (8k + 1) = 0, is a perfect square.

Find the least positive value of k for which the equation x² + kx + 4 = 0 has real roots. 

Find the values of k for which the quadratic equation 

\[\left( 3k + 1 \right) x^2 + 2\left( k + 1 \right)x + 1 = 0\] has equal roots. Also, find the roots.

Find the values of p for which the quadratic equation 

If −5 is a root of the quadratic equation\[2 x^2 + px - 15 = 0\] and the quadratic equation \[p( x^2 + x) + k = 0\] has equal roots, find the value of k.

If 2 is a root of the quadratic equation \[3 x^2 + px - 8 = 0\] and the quadratic equation \[4 x^2 - 2px + k = 0\]  has equal roots, find the value of k.

If 1 is a root of the quadratic equation \[3 x^2 + ax - 2 = 0\] and the quadratic equation \[a( x^2 + 6x) - b = 0\] has equal roots, find the value of b.

Find the value of p for which the quadratic equation 

\[\left( p + 1 \right) x^2 - 6(p + 1)x + 3(p + 9) = 0, p \neq - 1\] has equal roots. Hence, find the roots of the equation.

Disclaimer: There is a misprinting in the given question. In the question 'q' is printed instead of 9.

Determine the nature of the roots of the following quadratic equation:

(x - 2a)(x - 2b) = 4ab

Determine the nature of the roots of the following quadratic equation:

9a²b²x² - 24abcdx + 16c²d² = 0

Determine the nature of the roots of the following quadratic equation:

2(a² + b²)x² + 2(a + b)x + 1 = 0

Determine the nature of the roots of the following quadratic equation:

(b + c)x² - (a + b + c)x + a = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

x² - kx + 9 = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x² + kx + 2 = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

4x² - 3kx + 1 = 0

In the following determine the set of values of k for which the given quadratic equation has real roots:

2x² + kx - 4 = 0

If the roots of the equation (b - c)x² + (c - a)x + (a - b) = 0 are equal, then prove that 2b = a + c.

If the roots of the equation (a² + b²)x² − 2 (ac + bd)x + (c² + d²) = 0 are equal, prove that `a/b=c/d`.

If the roots of the equations ax² + 2bx + c = 0 and `bx^2-2sqrt(ac)x+b = 0` are simultaneously real, then prove that b² = ac.

If p, q are real and p ≠ q, then show that the roots of the equation (p − q) x² + 5(p + q) x− 2(p − q) = 0 are real and unequal.

If the roots of the equation (c² – ab) x² – 2 (a² – bc) x + b² – ac = 0 in x are equal, then show that either a = 0 or a³ + b³ + c³ = 3abc

Show that the equation 2(a² + b²)x² + 2(a + b)x + 1 = 0 has no real roots, when a ≠ b.

Prove that both the roots of the equation (x - a)(x - b) +(x - b)(x - c)+ (x - c)(x - a) = 0 are real but they are equal only when a = b = c.

If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations ax² + bx + c = 0 and -ax² + bx + c = 0 has real roots.

If the equation \[\left( 1 + m^2 \right) x^2 + 2 mcx + \left( c^2 - a^2 \right) = 0\] has equal roots, prove that c² = a²(1 + m²).

Find the consecutive numbers whose squares have the sum 85.

Divide 29 into two parts so that the sum of the squares of the parts is 425.

Two squares have sides x cm and (x + 4) cm. The sum of this areas is 656 cm². Find the sides of the squares. 

The sum of two numbers is 48 and their product is 432. Find the numbers?

If an integer is added to its square, the sum is 90. Find the integer with the help of quadratic equation.

Find the whole numbers which when decreased by 20 is equal to 69 times the reciprocal of the members.

Find the two consecutive natural numbers whose product is 20.

The sum of the squares of the two consecutive odd positive integers as 394. Find them.

The sum of two numbers is 8 and 15 times the sum of their reciprocals is also 8. Find the numbers.

The sum of a numbers and its positive square root is 6/25. Find the numbers.

The sum of a number and its square is 63/4. Find the numbers.

There are three consecutive integers such that the square of the first increased by the product of the first increased by the product of the others the two gives 154. What are the integers?

The product of two successive integral multiples of 5 is 300. Determine the multiples.

The sum of the squares of two numbers as 233 and one of the numbers as 3 less than twice the other number find the numbers.

Find the consecutive even integers whose squares have the sum 340.

The difference of two numbers is 4. If the difference of their reciprocals is 4/21. Find the numbers.

Let us find two natural numbers which differ by 3 and whose squares have the sum 117.

The sum of the squares of three consecutive natural numbers as 149. Find the numbers

Sum of two numbers is 16. The sum of their reciprocals is 1/3. Find the numbers.

Determine two consecutive multiples of 3, whose product is 270.

The sum of a number and its reciprocal is 17/4. Find the number.

A two-digit number is such that the products of its digits is 8. When 18 is subtracted from the number, the digits interchange their places. Find the number?

A two digits number is such that the product of the digits is 12. When 36 is added to the number, the digits inter change their places determine the number.

A two digit number is such that the product of the digits is 16. When 54 is subtracted from the number the digits are interchanged. Find the number

Two numbers differ by 3 and their product is 504. Find the number

Two number differ by 4 and their product is 192. Find the numbers?

A two digit number is 4 times the sum of its digits and twice the product of its digits. Find the number.

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.

The sum of two numbers is 18. The sum of their reciprocals is 1/4. Find the numbers.

The sum of two number a and b is 15, and the sum of their reciprocals `1/a` and `1/b` is 3/10. Find the numbers a and b.

The sum of two numbers is 9. The sum of their reciprocals is 1/2. Find the numbers.

Three consecutive positive integers are such that the sum of the square of the first and the product of other two is 46, find the integers.

The difference of squares of two number is 88. If the larger number is 5 less than twice the smaller number, then find the two numbers.

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.

The difference of two natural numbers is 3 and the difference of their reciprocals is  \[\frac{3}{28}\].Find the numbers.

The numerator of a fraction is 3 less than the denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and the original fraction is  \[\frac{29}{20}\].Find the original fraction.

The speed of a boat in still water is 8 km/hr. It can go 15 km upstream and 22 km downstream in 5 hours. Find the speed of the stream.

A passenger train takes 3 hours less for a journey of 360 km, if its speed is increased by 10 km/hr from its usual speed. What is the usual speed?

A fast train takes one hour less than a slow train for a journey of 200 km. If the speed of the slow train is 10 km/hr less than that of the fast train, find the speed of the two trains.

A passenger train takes one hour less for a journey of 150 km if its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train.

The time taken by a person to cover 150 km was 2.5 hrs more than the time taken in the return journey. If he returned at a speed of 10 km/hr more than the speed of going, what was the speed per hour in each direction?

A plane left 40 minutes late due to bad weather and in order to reach its destination, 1600 km away in time, it had to increase its speed by 400 km/hr from its usual speed. Find the usual speed of the plane.

An aeroplane take 1 hour less for a journey of 1200 km if its speed is increased by 100 km/hr from its usual speed. Find its usual speed.

A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train.

A train travels at a certain average speed for a distance 63 km and then travels a distance of 72 km at an average speed of 6 km/hr more than the original speed, If it takes 3 hours to complete total journey, what is its original average speed?

A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hour more, it would have taken 30 minutes less for a journey. Find the original speed of the train.

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.

An aeroplane left 50 minutes later than its scheduled time, and in order to reach the destination, 1250 km away, in time, it had to increase its speed by 250 km/hr from its usual speed. Find its usual speed.

Ashu is x years old while his mother Mrs Veena is x² years old. Five years hence Mrs Veena will be three times old as Ashu. Find their present ages.

The sum of ages of a man and his son is 45 years. Five years ago, the product of their ages was four times the man's age at the time. Find their present ages.

The product of Shikha's age five years ago and her age 8 years later is 30, her age at both times being given in years. Find her present age.

The product of Ramu's age (in years) five years ago and his age (in years) nice years later is 15. Determine Ramu's present age.

Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

A girls is twice as old as her sister. Four years hence, the product of their ages (in years) will be 160. Find their present ages.

The sum of the reciprocals of Rehman's ages, (in years) 3 years ago and 5 years from now is 1/3. Find his present age.

The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. Find the lengths of these sides.

The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

The hypotenuse of a right triangle is `3sqrt10`. If the smaller leg is tripled and the longer leg doubled, new hypotenuse wll be `9sqrt5`. How long are the legs of the triangle?

A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?

The perimeter of a rectangular field is 82 m and its area is 400 m². Find the breadth of the rectangle.

The length of a hall is 5 m more than its breadth. If the area of the floor of the hall is 84 m², what are the length and breadth of the hall?

Two squares have sides x cm and (x + 4) cm. The sum of this areas is 656 cm². Find the sides of the squares. 

The area of a right angled triangle is 165 m². Determine its base and altitude if the latter exceeds the former by 7 m.

Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m²? If so, find its length and breadth.

Is it possible to design a rectangular park of perimeter 80 and area 400 m²? If so find its length and breadth.

Sum of the areas of two squares is 640 m². If the difference of their perimeters is 64 m. Find the sides of the two squares.

Sum of the area of two squares is 400 cm². If the difference of their perimeters is 16 cm, find the sides of two squares.

Represent the following situation in the form of a quadratic equation:

The area of a rectangular plot is 528 m². The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

A takes 10 days less than the time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days, find the time taken by B to finish the work.

If two pipes function simultaneously, a reservoir will be filled in 12 hours. One pipe fills the reservoir 10 hours faster than the other. How many hours will the second pipe take to fill the reservoir?

Two water taps together can fill a tank in `9 3/8`hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Two pipes running together can fill a tank in `11 1/9` minutes. If one pipe takes 5 minutes more than the other to fill the tank separately, find the time in which each pipe would fill the tank separately.

To fill a swimming pool two pipes are used. If the pipe of larger diameter used for 4 hours and the pipe of smaller diameter for 9 hours, only half of the pool can be filled. Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool?

A piece of cloth costs Rs. 35. If the piece were 4 m longer and each meter costs Rs. one less, the cost would remain unchanged. How long is the piece?

Some students planned a picnic. The budget for food was Rs. 480. But eight of these failed to go and thus the cost of food for each member increased by Rs. 10. How many students attended the picnic?

A dealer sells an article for Rs. 24 and gains as much percent as the cost price of the article. Find the cost price of the article.

Out of a group of swans, 7/2 times the square root of the total number are playing on the share of a pond. The two remaining ones are swinging in water. Find the total number of swans.

If the list price of a toy is reduced by Rs. 2, a person can buy 2 toys more for Rs. 360. Find the original price of the toy.

Rs. 9000 were divided equally among a certain number of persons. Had there been 20 more persons, each would have got Rs. 160 less. Find the original number of persons.

Some students planned a picnic. The budget for food was Rs. 500. But, 5 of them failed to go and thus the cost of food for each member increased by Rs. 5. How many students attended the picnic?

A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?

In a class test, the sum of the marks obtained by P in Mathematics and science is 28. Had he got 3 marks more in mathematics and 4 marks less in Science. The product of his marks would have been 180. Find his marks in two subjects.

In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.

If \[1 + \sqrt{2}\] is a root of a quadratic equation will rational coefficients, write its other root.

Write the condition to be satisfied for which equations ax² + 2bx + c = 0 and \[b x^2 - 2\sqrt{ac}x + b = 0\] have equal roots.

Write the set of value of k for which the quadratic equations has 2x² + kx − 8 = 0 has real roots.

Write a quadratic polynomial, sum of whose zeros is \[2\sqrt{3}\] and their product is 2.

Find the discriminant of the quadratic equation \[3\sqrt{3} x^2 + 10x + \sqrt{3} = 0\].

If \[x = - \frac{1}{2}\],is a solution of the quadratic equation \[3 x^2 + 2kx - 3 = 0\] ,find the value of k.

If the equation x² + 4x + k = 0 has real and distinct roots, then

If the equation x² − ax + 1 = 0 has two distinct roots, then

If the equation 9x² + 6kx + 4 = 0 has equal roots, then the roots are both equal to

If ax² + bx + c = 0 has equal roots, then c =

If the equation ax² + 2x + a = 0 has two distinct roots, if 

The positive value of k for which the equation x² + kx + 64 = 0 and x² − 8x + k = 0 will both have real roots, is

The value of \[\sqrt{6 + \sqrt{6 + \sqrt{6 +}}} . . . .\] is 

If 2 is a root of the equation x² + bx + 12 = 0 and the equation x² + bx + q = 0 has equal roots, then q =

If the equations \[\left( a^2 + b^2 \right) x^2 - 2\left( ac + bd \right)x + c^2 + d^2 = 0\] has equal roots, then

If the roots of the equations \[\left( a^2 + b^2 \right) x^2 - 2b\left( a + c \right)x + \left( b^2 + c^2 \right) = 0\] are equal, then

If the equation x² − bx + 1 = 0 does not possess real roots, then

If x = 1 is a common roots of the equations ax² + ax + 3 = 0 and x² + x + b = 0,  then ab =

If p and q are the roots of the equation x² − px + q = 0, then

If a and b can take values 1, 2, 3, 4. Then the number of the equations of the form ax² +bx + 1 = 0 having real roots is

The number of quadratic equations having real roots and which do not change by squaring their roots is

If \[\left( a^2 + b^2 \right) x^2 + 2\left( ab + bd \right)x + c^2 + d^2 = 0\] has no real roots, then

If the sum of the roots of the equation x² − x = λ(2x − 1) is zero, then λ =

If x = 1 is a common root of ax² + ax + 2 = 0 and x² + x + b = 0, then, ab =

The value of c for which the equation ax² + 2bx + c = 0 has equal roots is

If \[x^2 + k\left( 4x + k - 1 \right) + 2 = 0\] has equal roots, then k =

If the sum and product of the roots of the equation kx² + 6x + 4k = 0 are real, then k =

If sin α and cos α are the roots of the equations ax² + bx + c = 0, then b² =

If 2 is a root of the equation x² + ax + 12 = 0 and the quadratic equation x² + ax + q = 0 has equal roots, then q =

If the sum of the roots of the equation \[x^2 - \left( k + 6 \right)x + 2\left( 2k - 1 \right) = 0\] is equal to half of their product, then k =

If a and b are roots of the equation x² + ax + b = 0, then a + b =

A quadratic equation whose one root is 2 and the sum of whose roots is zero, is ______.

If one of the equation ax² + bx + c = 0 is three times times the other, then b² : ac =

If one root the equation 2x² + kx + 4 = 0 is 2, then the other root is

If one of the equation x² + ax + 3 = 0 is 1, then its other root is

If one root of the equation 4x² − 2x + (λ − 4) = 0 be the reciprocal of the other, then λ =

If y = 1 is a common root of the equations \[a y^2 + ay + 3 = 0 \text { and } y^2 + y + b = 0\], then ab equals

The values of k for which the quadratic equation  \[16 x^2 + 4kx + 9 = 0\]  has real and equal roots are